The way you use settings API i deprecated since WP 2.7. Tere are also debug notices like Notice: Undefined index: q2w3_fixed_widget in ...\wp-content\plugins\q2w3-fixed-widget\q2w3-fixed-widget.php on line 59 because you should check if key q2w3_fixed_widget exists in instance table before you check it for vaue. (WP_DEBUG recommended)
Settings related to tablets and also checkboxes "Fixed widget" for individual widgets are NOT stored in database. So, fresh installation of plugin (never installed before) simply does not work.